Crystallography & Symmetry
patterns derived from their groups
Concept

Fixed point — where it appears

The point an operation does not move, which for a rotation is its centre and is where its mark is drawn. Whether an operation has one at all is what separates a rotation from a screw and a mirror from a glide.

Named by 13 essays across 4 fields — each of them below, with the objects they name alongside it.

The 4₁ screw axis. 1 of the eleven screw axes a lattice permits, each drawn as the helix it is: 4₁. A turn of 2π/n followed by an advance of m/n of the repeat, so that n turns land exactly m cells along. 0 of those drawn are its own mirror image; the rest come in left- and right-handed pairs.

Turning and climbing at once

A rotation has a fixed point and a screw has none. That sounds like a small difference and it is the reason a space group is not a point group with extra letters, the reason two hundred and thirty is not seventy-three, and the reason a helix can be a crystal.

space-groups · Screws and glides Two half-turns make a translation. The half-turn about (0.25, 0.25) followed by the half-turn about (0.75, 0.5) is the translation by (1, 0.5) — twice the vector between the two centres, and not the vector itself. The open lens is a third centre, and it is not the midpoint of the two drawn: it is where the half-turn about the first lands when it is composed with one repeat vector of the lattice, which is half a repeat along. That is the step that puts two-fold centres on the half lattice and gives a p2 cell four inequivalent ones. Both the translation and the forced centre are computed from the operations and compared with the construction in exact rational arithmetic.

Where the product is

Composing two symmetries lands on a third — and the third one is somewhere. Two half-turns make a translation by twice the distance between their centres, and that single fact puts the lattice into a pattern before anybody chooses one.

operations · Composition What each group leaves distinct. The number of genuinely different ways of putting 2 species on the cells of a 4 × 4 block, for 9 plane groups. Every row starts from the same 65,536 arrangements; what differs is the group identifying them. Each count is Burnside's average of fixed points, and each was required to divide exactly by its group's order.

Counting what a group cannot tell apart

Sixty-five thousand ways of putting two species on sixteen sites; eight hundred and five structures. The difference between those numbers is not a division, because the symmetric arrangements have short orbits — and the count that gets it right is an average of fixed points.

operations · Counting glide: 3 mirrors. A glide of the plane, drawn together with the mirrors it is a product of. The first shape is the motif; the pale ones are what each mirror in turn produces; the last is the image the motion itself gives. There are 3 mirrors, which is the smallest number that can produce this kind of motion, and their product was formed and compared with the motion before the figure was drawn.

Three reflections, and never four

Every motion of the plane is a product of mirrors, and the number needed is never more than three. That count is not a curiosity about mirrors — it is the classification of the four motions written as an integer, with the parity of the number deciding handedness and the geometry of the last two mirrors deciding everything else.

operations · Composition P4_1: a screw of 90°. One operation of P4_1, reduced to Chasles' three numbers: an axis, an angle of 90°, and a pitch of 1.25 along it. The points are the orbit of one position under repeated application, which climbs because the pitch is not zero — and it is not zero for any choice of origin, which is what makes this a screw rather than a rotation. It needs 4 mirrors, and their product was checked against the operation before this was drawn.

Every motion of space is a screw

A rigid motion of space that preserves handedness turns about some axis and slides along that same axis, and there is nothing else it can do. Rotations and translations are the two ends of that one description, the axis and the pitch are computed rather than recognised, and the operations a space group is made of stop being a list of kinds.

operations · What symmetry is Averaging a metric over the group. The 3 pale ellipses are the unit circle carried by each element of a finite group of rational matrices — none of them a rotation, because the group has been skewed out of the orthogonal ones on purpose. Their average is the heavy ellipse, and it is invariant: MᵀAM = A for every element, exactly, in rational arithmetic. So a finite group of matrices is always a group of isometries of some inner product, and every question about how large such a group can be becomes a question about the symmetries of an ellipse. The space of invariant forms here is 1-dimensional, so up to scale the average is the only one.

The average that makes it finite

Two arguments every classification leans on are usually assumed rather than made: that a finite group of motions fixes a point, and that a finite group of integer matrices preserves a metric. They are the same trick — average over the group — and the trick fails exactly where it should.

restriction · Finiteness p1 folds into a torus. The cell of p1 with its edges marked as the group joins them: both pairs by a plain translation, both arrows the same way round. Gluing top to bottom gives a tube and gluing its ends gives a torus. Nothing in p1 holds a point still, so the surface has no marked points and its first homology is two copies of the integers.

The two that fold into a surface

Fold a wallpaper pattern along its own symmetries and what is left is usually a shape with corners and edges nobody drew. For two of the seventeen it is a plain surface with no marks on it at all — a torus and a Klein bottle — and which two is decided by a single question asked of every operation.

classification · Flat space Ten ways for space to be flat. The thirteen groups, with each mirror-image pair counted once, because a shape and its mirror image are the same shape. 3 of the ten arrive that way — the three-fold, four-fold and six-fold screws, which are the enantiomorphic pairs this collection already counts among the two hundred and thirty. Six of the ten are orientable and four are one-sided.

Ten ways for space to be flat

Thirteen of the two hundred and thirty space groups hold no point still, and folding space along one of them gives a shape with no curvature anywhere. There are ten such shapes, not thirteen, and the difference is the same eleven pairs that separate 230 from 219.

space-groups · Flat space Four of the seventeen have a centre, and they are the four with no rotation. For each plane group: the order of its point group, how many of its operations are rotations, the lattice vectors every operation of the point group fixes, and the centre those vectors make. A central element must commute with every translation, which forces its linear part to be the identity — so the centre is a group of translations, and a translation is central exactly when the point group leaves it alone. A rotation leaves nothing alone but zero.

The four groups with a centre

An element that commutes with everything has to commute with every translation, and that forces its linear part to be the identity. So the centre of a plane group is a group of translations — the ones its point group leaves alone — and a rotation leaves nothing alone but zero. Four of the seventeen have a centre and thirteen have nothing at all.

operations · Composition Two turns and their undoing leave a slide. A turn g by 90° about the point c and a turn h by 60° about d. The marked point p is carried back 60° about d, back 90° about c, forward 60° about d and forward 90° about c, and does not return: it arrives displaced by a vector of length 2.371, which is 4·sin 45°·sin 30°·|c − d|. Two other points put through the same four motions move by the same vector, drawn beside them, because the commutator g h g⁻¹ h⁻¹ of two rotations of the plane is a translation — (I − A)(I − B)(c − d) exactly — whatever the angles and the centres.

What forces a lattice

Every enumeration here starts from a lattice of translations, and the lattice is usually taken as given. It need not be. A group of motions that is discrete, and leaves no point far from an orbit, has to contain one — in the plane by an argument four lines long, each line a picture, and in space by an inequality whose threshold turns out to be the six-fold rotation.

restriction · Finiteness Three conditions, and a near-miss for each. Zassenhaus's characterisation asks a group for a normal subgroup that is free abelian of finite rank, of finite index, and maximal among the group's abelian subgroups. Four groups against those three clauses. The free group on two letters has no non-trivial abelian normal subgroup at all; the discrete Heisenberg group has one that is free abelian of rank two and maximal abelian, and its index is infinite; ℤ² × ℤ/2 has a free abelian normal subgroup of index two, and the maximal one has torsion in it. Each fails a different clause, which is what shows no clause is redundant. The infinite dihedral group passes and is crystallographic in one dimension.

Which groups a crystal could have

Bieberbach's theorem is a statement about a group acting: discrete, no point far from an orbit. Zassenhaus turned it round into a statement a group can satisfy on its own — a maximal abelian normal subgroup, free of finite rank, of finite index — and each of those three clauses is kept out of redundancy by a group that fails it and nothing else.

restriction · Finiteness An orbit on a parabola, discrete and cocompact. The images of the origin under the group generated by two commuting affine maps of the plane: A slides one step along x and lifts y by the x it started at plus a half, and B is the translation by one in y. The images are the points with whole-number first coordinate and second coordinate a whole number above half the square of it, so the large dots lie on the dashed parabola and the small ones are the rest of the orbit. No two distinct images come closer than 1.000, and no point of the square between the axes lies farther than 0.610 from one — so the action is discrete and its quotient is compact, which is exactly what Bieberbach's first theorem asks for.

Straight lines, and no distances

Every finiteness met so far rests on the motions preserving a metric, because the trick that produces one is an average and an average needs something to average over. Keep the straight lines and drop the distances, and Bieberbach's first theorem is false in the plane — by an example two lines long, whose group is the plane's own translations and whose translations have rank one.

restriction · Finiteness A screw out of two rotations that have none. Four pairs of located rotations of space, composed, with the result read back as a screw: its angle, and its pitch, which is the part of its translation lying along its own axis. Axes that meet give a rotation and no translation at all, because the point where they meet is fixed by both. Parallel half-turns give a translation. Skew axes give a screw — a motion with a translation in it, out of two motions with none — and the translation is twice the distance between the two axes. Nothing in either factor moves anything along the product's axis, and the product does.

The axis a product lies on

Two rotations of space about axes that do not meet compose to a screw — a motion with a translation in it, out of two that have none. The translation is twice the distance between the axes and the angle twice the angle between them, and the screw's own axis is not somewhere arbitrary: it lies on the two axes' common perpendicular, at a place the arithmetic gives.

operations · Composition

Named alongside it

The objects these essays reach for when they reach for this one.

Lattice translationCompositionFinite groupOrbitScrew axisSymmetry operationTranslation groupDiscretenessIntrinsic translationAbelianisationCartan dieudonneCommutator

All concepts